# Full text of "Notes on Groups and Geometry, 1978-1986"

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Nates on Groups and Geometry, 19 78" 1986 by Steven H. Cullinane Typewritten notes coHected in a 40-page PDF document. CONTENTS 01 1978-??-?? "An inrariance of symmetry*' Research announcement- OS 1978-10-?? "Symmetry invariance in a diamond ring," AXIS abstract received Oct.. 31, 1978. 03 1978-12-?? "Orthogonality of Latin squares viewed as as skewness of lines" 04 1981-11-05 "Patterns invariant, modulo rigid motions, under groups of discontinuous transformations- two examples" 05 1981-12-24 "Solid symmetry"-- "Motions of each cell induce motions of the entire pattern." 06 1982-05-12 "Map systems" Definition. 07 1982-06-12 "I nscapes" Definition. 08 1982-09-12 "A symplectic array" 09 1982-09-22 "Inscapes II" Generalized definition. 10 1982-12-27 "Group scores" Definition. 11 1983-05-31 "Decompositions of group enveloping algebras* 1 12 1983-06-21 "An invariance of symmetry* 1 Group actions on a 4x4x4 cube. 13 1983-08-04 "Group identity algebras" Definition. 14 1982-08-26 "Transformations over a bridge" Definition and problem. 15 1983-10-01 "Portrait of O" Action of the octahedral group. 16 1983-10-16 "Study of O" Group actions on two-color cubes. 17 1983-11-08 " Compound groups* B efinition and problem. 18 1983-11-10 "Group compounds" Definition and problem. 19 1983-11-27 "Table groups" Definition and problem. 20 1984-01-05 "Line ar op er at or s in ge ometric function spaces" Definition, theorem, problem. 21 1984-09-15 "Diamonds and whirls" Blocks illustrate group actions. 22 1984-09-25 "Affine groups on small binary spaces" Theorem. 23 1985-03-26 "Visualizing GL(2,pV' Example on a 3x3 array. 24 1985-04-05 "GL(2,3) actions on a cube" 25 1985-04-05 "Group actions on partitions" Definition and problem. 26 1985-04-28 "Generating the octad generator" 27 1985-08-22 "Symmetry invariance under M12-" Theorem. 28 1985-11-17 "A group bridge" Definition and problem. 29 1985-12-11 "Dynamic and algebraic compatibility of groups" Problems. 30 1986-01-11 " Ge ometry of p artitions II " A foray into analysis. 31 1986-02-04 "Inscapes III: PG(2,4) from PG(.3,2)" 32 1986-02-20 "The relativity problem in finite geometry" 33 1986-03-31 "Group topologies" Definition and problems. 34 1986-04-26 "Picturing the smallest projective 3-space" 35 1986-05-08 "A linear complex related to M2 4" 36 1986-05-26 "The 2-subsets of a 6-set are the points of a PG(3,2)" 37 1986-06-26 "21 projective partitions" A 6-set model of PG(2,4). 38 1 9 8 6 - 06- 1 1 "An out er automorp hism of S 6 relate d to M24" 39 1986-07-03 " Picturing outer automorp hisms of S 6 " 40 1986-07-11 "Inscapes IV- Inner and outer group actions" Web page URL,- http7;finitegeomet:iyorg/sc/gen/typednotes.html AH 1HVAHIANCE 07 SYMMETRY m 3TEVEJI E. OUIilHANE (Written sometime in 1978.) We present a simple, surprising, and beautiful combinatorial Invariants or geometric symmetry, *0 an algebraic setting* DEFINITION, A delta trgn_afoim of a square array over a U.-set is any pattern obtained from the array by a 1-to-l substitution or the four diagonally-divided t*C-color unit equates for tha if-sot elemonts. EXAMPLES. ffl-fflHSSSH .... PHEOSSIL. Every delta transform of the Klein group table has ordinary or col or- interchange symmetry, and remains symmetric under the group Qt Of y22. % $hfo transformations generated by combining permutation a of rows and columns with permutations of -quadrants* EXAMPLE. ■fl ^ $ o ^ M PROOF (Sketch). The Klein group ia the additive group of CF(^); tbls suggests t*e regard the group's table T as a matrix over that field. So regarded, T is a linear combination of three (0^1 ^matrices that indicate the locations, lr. T, of the £-subsets of field elements. The structural symmetry of these matrices accounts for the ayrnnatpy of the delta transforms of T^ and is Invariant ■under C All delta transforms of the ty? matrices in the algebra generated by the images of T under fi are symmetric; there are many such algebras. THEOREM. If l<m±n z + 2 t there is an algebra of Ij.™ 2n 7. 2ti matrices over GF(lj) with all delta transforms syrueetrie* An Induction proof constructs sets of basis matrices that yield the desired sysjuetry and ensure closure under multipll eats ion. REFERENCE S. H. Cullinane, Diamond theory (preprint)* Notices of the American Mathematical Society, February 1979! Issue 192, Volume 26, Number 2., pages A-193-A-194: Tyr-JOT H SttVtb """ ■*- CulllnaoE Syrngtrr iflv*rl*Bet in ■ JImcbI rtng . >raltaliHTT T»»0« . Hi riiiu ibi r«ur-4tn*and fliUTt D •■ * MU arm «f 1-eolar tilt* inch •■ Q. in c h tha [roup ef 312, Its ptnutatLsM cf rhtia 14 dlta t*t\*ttt*i fcy urbUHrlh "Hlm tivtim (•■rauiatlroa of rwi iemL of teluaaa trlch ranAoaj partntatloni *( lb* four 1*1 quad rial t . Etfarj -C-tHui e[ P bat ho* ordinal? o* color- Lrn arching* aymeerr. EUHTLTi IlmllD-fiS^ fk**T# |( C u t rroddtt of tun d la j tlnC J~*T«I*»). ■at* thlt 1% (via rotltlMal IIiltif-inEtfTClUitf* araPttty Ilia llui af Iha fjaiiui jHj - T)f i| aviilml, MXUUi fl la lawrfM* <» tha aftld* ir+ua an V 4 {GT{l))i Tha » atTuetwm cf thr 140-1 W* G-JauLaaai pf p pti laeaorphle to tha 35 llmra In the J-dLuul-Hul frajaetiva tf4tt Ottr CT{l)i i>r|Ka|*ii*tltr al niMiuiii vutttrnfmi* 14 tttwitii «r 1t»». ill can dattaa naai and pioa'aco *• ibai Ilvl G-faaiaa «f ■> |ta*tat* an Uaal (Jbfli* pattaina ctuKctarlaad If all tarliemal nr vartUal Vina" **lnf >ni I nt mh pi *■() : f a tlnj nC * , Dl* rfaarlTli pattern!, Th*Tl ll H Inflrill* ruUr of HMK "Jlunhd" rtnma. Hflajorphle to rldii u( ■■trlraa ov» T »f*l- (Jtacalnd October 31, 1T7B. ! STEVElf H. CTLLBKH5 . Orthogonality Of Latin squares flawed fl.a a]cow»CBE of l ingj^. Shown below is a way to urcibcd tho six ordcr-U Latin squares that iiavp orthogonal Latin mates in a oat of 35 arrays so that orthogonality in the sot Of arrays corregpondfl to skeTmeaa in the act of 35 Una a of ?G{3»2}, Each array yioldB a. 3-set of diagrams that Show the Urea separating conrplamantary E-aiiDaeta of {0,,5»2,3} * each diagranq is the symmetric difference of the other two. The 3-aata of dlagraias correspond to tha lines dT FG(3,2}. Two arrays ara orthogonal Iff ■ their 3-aots or diagrams are disjoint., i.e. iff the corresponding lines of FGO r 2) eto aitcm. Thla is a new way of -slewing orth.CE on allty ° r Latin aqiiaraa, quite different frora their relationship to projectivo plane a. PHDBLEW: To what extent can this result be general laud 7 {Doe. 19?S ) OH} I»1X 3 I I B I O J i 11 I a ffl — I. i t J 41 I 31 I a 1 O 3 1 OI15 13 C I — ]—f T7 ? i '< 3 5*10 I 3 1 till Oll!3 3 a ■ a 13 8 I («ll EL m n il OIIO 1 a I till 1 O L O 1IJ.S am I 1 3 e I i S o n o 13-Jl ill! 1 D D 1 oixl 43 a I 7- "So I t> I a I Jill a t e 1 1 Q I 1 o 1 a t513 3131 o | jt-3 ^-^e. 1 a a i i 1113 o a » 1 D 1 o I I 1?13 f #g I t I I a o o I 1-3 3ll» o i I. ■> 3Ho [ 1 ffl .:. i i t! 1 C £3 J 11J1 Till a I <L1 )t|o o lis = lio e M o 11"- » 3 i^- »l IQ 1JJ1 D I I < I o o ■312. 2 ■i ( ■- 3 i o?l Q I 43 1 »3 i Ci I o I I a | □ JI31 eiM old ■i 3-3-3 1321 "I ' J . L : . o 1*3 Jllo a d^qp a III' 7.41 1- 3J35 1131 lOjl .■' I 1 c. 1131 ; it 1 C 5 1 i Del I H3> ay- 3? 00 I I qo I 1 tftil Ji?1 il!i Q 1 U (pl!3 1 o3 £ do 1 1 3312. o I 4-3 13 o 1 »lll 2.1 a I 1 1 UJj - - 1 ■ 1 — 1 j ill Steven 11. Cull, inane Patterns .i nvar iian; i, modulo rigid motions, uncer c?roat>s of aiEcontinuog g transfoiTiittons: lyo examples , Rtseatch i-i&fc<s. November 5, 1981.. In fig- 1, rigid motions of each cell in a pattern induce rigitf motions of tne entire pattern. In fig. 3, permutations of tells proc^ce various sectAOival vtevs of the sarna (modulo field motions) infinite plane pattern. These permutations are derived, as in fig. 2, item motions of a cub*. Fifi. 1 ft* DC £ F H & H G, H o F 6 & a C P £ ft & F H E, t & &F A £ "0 H E. F A3 H & It c B A K & 3 ft F £ ft F "VI f> t F ft £ f a £ A. A.E. &F prt C5 B A f £ c D C H ftpn E H F& B c ftp f-1 £ A G-c- PlS- ? aa is ra ei k k i .3 ta rs ra K as o c « O A a £ F H 3> £<: EA P & F £ I* AE FIS- 3 en S, Cull inane jty^inBtrv . Expository note. December 24, L9fll. In pattern A, motions of each cell induce motions of the entire pattern; li.fcew3.se in B. B Steven H- Cullinane Mao systems . Query, May 12, 1982. Definition: Suppose every map d) into a given ring « can be vritten aa <f K ^ £;(*,.' 4*) . inhere N is a fixed positive integer, the e, are fixed elements of M, and the 0^ are fixed functions from M to P, b proper subset of M. Let A » {« *it > , = (* 1 . ... , IgJ-. The quintuple (M, A, P r 0, B) is a map system . Example: Using hexadecimal labels for the elements of GFU6] , l«t {H. A, I?, 0, H) - (GF(U}, A, ^O^B,^, f6, B, P) , 3). where the functions in A are specified by giving inverse images: 10,1,3,21— iOl l0,4,C,Bl~i0l {0,5,*,A\-(Gt \ (C»D,F,E)~[B) (3.7,F,lM~»m U , 6.C9 J~l 9 } ifl^,B,A)-{9>, {2.6,E,A)*tEl, [2.7.D,8W7} / (*„«», *3 > {Multiplication in gf(1$J U here defined via the irreducible polynomial m 4 + X + l.J Remarks: Harmonic analysis allows a complicated map to be broken down into, or built Up from, simpler maps. Map systems are a different means to the same *nd. Query: What is known about such systems? Steven H. (Jul limine Inac&pfls . Query. June 12, l^ftS. Definition: Let R be an n-aiy symmetric relation on a set of t subsets of a t-eet, iftiere n< t -ut, for positive integers n,t,u,v. Represent eaoh Of the t subsets toy the l's In a nit array u^ over GF(£), There li l£ t. An In scape of R la a ujct array A of tho a^ flttah. that R is true for n of the a^_ {that is, for the subsets rapres anted by these a^) If and only if the arrangement of the a^ within A Is the same aa the arrangement of the l's In some a^ <£ . Examples: (light and dark represent O's and l r fl.) nun deeh HSEBBH HCBO -COCO ESDHn Remark a : Inscapes are useful for visualising relations in certain finite geometries . The above examples, for instance, illustrate relations among the 15 hyperplanea of ?G(3*2) and among the l£ lines fliad undar a particular symplsitlc polarity of F0(3i-2). Quepy: What is knoim about coaibinatorlal systems of trie sort? Note: For some other properties of the aj In the second example, sea B.F. Assaua, Jr. and J.E, Not5.Ho Sard.1, "Oari^ralizad 3fceiner ey stems of type 3-{*o", \ I).j6j jl] 1l » Finite Qeometrlea and Designs , London Hath. 3oe. Lecture Note Series h$ { Cambridge tinXv. Frese, Cambridge 198l}, pp. lb-Zl. Steven H* Cull in one A Bjmploctic array . Research not a. September 12, 19fl2. Tho 11x11 array below Is farmed by adding (light = G, dark = 1, 1+1- 0) the 10 nonempty squares In the first column to the 10 non&iupty squares in tho flrnt row. These squares represent tho 10 pairs af lines interchanged mid or A particular oyuplectlc polarity Of PG{}*£}» The array Id of interest for several reasons ; 1) It Sorven to Illustrate an eleavoatary, but useful, way of construct Lng a complicated combinatorial object from simpler objects: make an addition table. [ Closure Is not asaentlal.) 2) Properties of array o thun for mod as addition tables may be of some Use in tho study Of 10x10 Latin squaree. 3 J Site* each of the 121 Jp4* squares below ro pre a on t s a set of points In a finite projective space* the array may aorva to illustrate or to suggest properties of such 3pacoa. ncBBEannaHH □GHKHQCaSHaiH HBHtzsaaaaHE] 8 St o Ten H* Culiinnna In scapes II . Query, September £2 $ 1962. Given a set X of points, certain families- of subnets of X nnay havB, as families , Home property s» (Example: the families of spheres that are concentric, ) It may be that i?o- can associate to each point of X & subnet of X, via an Injection f:X-^2 , in ouch a way that the f-itaage, in turn, of this subset of X (i.e., the family of f- Imago a of it a points) is in fact one of the families of subsets of X that have property a. If the map f gives rise in this way to the sat & of a 11^ such s-fainill&B, we can write. In a cryptic but concise way, S=f£f(X)), and say that f la an In a cape of Si Query: "What knonn reaulta can he stated, after the appropriate definition of 3, in the form "There exists an inacape of S fl F Addendum of Oct. 10, 1982. 4 more precise definition: Let X be a non-empty set . L&1 ; P{X) denote the set of all subs ats of X. Let ScP(PUJ), Suppcw se there exists an injection f: X-*P(X> such that, for any «-« pCp(x)), <rta if and only if 3x«X such that o'=f[f(i}J a {t(7)\ jtfW} . Th« n f is an Inacape of 3. This notion s rises naturally in studying the action of a symnlectie polarity in a projective space , Oaa of course wonders whether It has arisen previously in any other context. $t*ven H. Cullinano Groyp .ac.orea i Jfroblera. December 27* 1982. Definition: Lat G, be a finite permutation group. Represent <k >9 t group of permutation matrices over GF(2), febe tno-sl8TOBHt GflLoia field, and let V^ = ^t^ * denote too enveloping algebra* of £k, Suppoao there e*lst aubelgebr&a V 2 and V, of Y^ such that a] V 3 is * tranevoraal of the additive cossts of V 2 in 7j ■ and b) 7] li tno enveloping algebra of a subgroup G^ of G^. Then (& 1P V lf T 2 , 7^, G 2 J is a KroUo faQM> Example (Light and dark represent O's and l's): D0HDDDQD HMBBnoaca HacacBHcn EHHHMHHH Thla array A of matrices is a group acore In whicb 7^ A, 7, = the first row of A, 7, = the first aoluawi of A, and Q = tha permutation matrices in the firat column, Hote that In the example neither 7 nor 7^ ia an Ideal of 7^ Problem: YEiat group score a exist? * 3o e Flath, AWS abstract 797-17-&6. the related 7^7-2*>-130> **li H. ffoyl, Tne elasalcal groups. 2nd ed. (191+6} P Princeton, p, 79 10 Staven H. Culllnuns . -- _- fl Decompositions of group enveloping algebras . Query, Hay 31, iyo.j. No tat 1 on i Lot G be an abstract group , E a subgroup of 0. Let /t:ft+M be a representation of as a group tf of invortible endcunorphlsme of an R-module V, idler* R in a conuutativo ring with unity, and let B*ye(E). Denota the enveloping algebra of It (i.e., the R-linear oloaur* of M) by S{U\, or, rare explicitly, by (E{M),+ , ■ )• For a,b In £f(U) lot ta,b]^ a*b - b*a, and denote the resulting Lie algebra by (E(M),+ , CJ J. Query i 1, How can we relate decompositions of (E(M),+ , - ) to the structure of G? (In particular, when can wo write StM) as * direct sub where A is a aubalgebra of E(M)?) 2, How can we relate decompositions of {E(X) f +,CJ ), whore T - U or N, to the structure of G, when M Is nonabellan? (In particular, ho* are tfc -e Levi direct stud decomposltlona* E(lE) = H(U) + L(H) and £(V)=a(II)+ L(N), where R(X) Is thB radical of {E(X],+ , £3 ) and L(X) = E(X)/fl{X) is a sonrialnrple Lie subalgebra of (E(X) ,+- , C3 ) f related to the structure of &?) 3, How should we restrict the natures of Q, H, p , It, K, and V in order to answer (1) or (2) above? * See h, Ia Hal'cev, On semisiji^le subgroups of Lie groups (19Ui) • AMS Translations, series 1, volume 9 09e2) . 11 Steven H. CuHInaiie An Invariants of flyimsetry. 3s search note. June 21, 1983. B Theorem: There exists a triply trana3.-tl.Tr* group & of 1,2^0 ,157, k^k-i^k^ permutatiotia of the 6fy subcubea of B such that every G- image of B has a rigid-motion symrastry, (The marking on each eubeube of B is Identical; Bach la symmetric tinflar reflection in itfi coivfcay. ) Proof (sketch): ffa label the- 6k noils of B with the points of the afflne 6-apace A over (JF(2) In such a way that each hyperplasia of A is left Invariant or ±3 carried to ita complement, under a group C of S rigid motions generated by reflections in old-planes or B, We then dafina the group 0- &.Q tha group of afflne transforation a of A, Under Gj ae under Cj the set of bypsTplanes Bind. hyp erplane-cojiipl amenta is left invariant. This aynmietry of hyperplanea la then fairly easily shown to underlie the remarkable inTarianea of sjaaustry of B* (For a geometrically natural way to generate & aee £113 abstract 79T-A37.) 12 Stfliron H» Culllnane Group Identity algebran. Problem. August t|_, 1*^3. Definition: Let (3, »} and (S, o ) be groups with the jwu« sat S of element' symbol a but with different group tables. If there is at least one nlgabraic Identity I expressing a nontrlTial relationship between * and ° then fS > *, o } ia a sort of algebra, which for lack of any other najna we call a group Identity algebra. Exajmple: Let 5 z {e,a,b,o'J and let * and * be the operations +- and * {or Juxtaposition) In the table e below. + a b c a e a b c a a • c b fa b c a a c c b a a ■ fi n b o B H b B a b a a fa b a a a c c A Q b The following identity I holds. V ijiijirt (e>ajb»e\j (txy)+ (™}) + (U + y)U+ w)} - (Us) + (yw)) + (U + i)(7+ *)) = (<xw) + (yz)> + (U* w)(j+*]>. The dual identity I 1 obtained fay interchanging + and * also holds,, (Note that in thia case I and 1' state that certain algebraic forma are Invariant under the action of the symmetric group on their lndetermlnates. ) Froblea: Are there infinitely many finite group identity algebraa? (Mote the word "nontrlTial" in the definition. } 13 + a a b c e o a. b e n ft e c b b b c e a = o b a o - l a a b c 6 • a "b o a. a b c e b b a e a c c e a b Steven H, Cull inane Transformations ever a bridge . Problem, August 16, 19B3« Lst [G, + , * J bo the algebra (In the sense of universal algebra) with underlying aet G^(o,a,b,c^ and operations as follows. (.Re follow the not at I o rial conventions of writing * as juxtaposition and letting * precede + to avoid a proliferation of parentheses,) {Such an algebra (G,# j*)j where (0, *) and 1<J,^) aro groups, we call a bridge . } Problem . Part _1 . what is the nature of the group T generated by permutations of Gxfl of the form t(p,q,r,ft): lx,y) -*■ (px + qj* rx+ sy) -+- (x»y) nhere F,q t rjS 4 T Note 1.1 . It appears that T Is Isomorphic to a subgroup of the group of regular affine trans formations of the afflne |j- apace over GF(2). Hote I.E . we can have tCp^q^r, a) =: t(t,u,T>w) where fp,q#r,s)^ (tl,UjT,wJ since, for instance* V(x*yH 6-xG, ajc+ay = cx+cy. Part 2 . Is there some reasonably simple algebraic expression over (0*+,') for t{p,q,r,s) a t(t,u,v,ir) T Hflte a.l . Hot every member of T can be written in the form t{p,q,r, s). Example: (tfe., c*o,e) ) ♦ 14 Steven H. CulUQftne Portrait of 0. Action of tlio octaiwar&l group on a diamond. Oct. 1 'Hj, n^XN virfiay pxsy^ *ft^<p asv^ «>cw MX+D yfc^V! VS?A» ^^2VKA«i>« IKD4^n^V^« ^d*** V2AJ5 OKO £7iYi& 5^^^ ♦IHH SVAC ><H» «%«(9 sz;y£> a^fS^ DHti Asayt<HKiR(S'« , a ■y*5^*I «»5P^ MQX4 5AVa^[K>tfVi^» »«y^i?«^ft I4MQ ^2AOy]«^^ ^V^Si ♦MXO *C^^^ >«<M *^K S2VA 2sv^ aiHi *fi^y^ c<<ki> »<b<p;v avsjs *<<s?s^ M + ax &**?«!& w>r««: tf**3> 2SAV ^^ID*M^«iy <T<>H ^K^tf VA32 ft?W^(!? t^>^< VffSA 04MI £^ft ATVifc^ **d>i& >C<<H 5JVA3 ♦DIMV^««^9l d"&«* 02>H AS5V XM4D »a«9 ^^^^ %«s>«* w<m> SAV7 nrn* fX&w ^y*c« SVEA fc**d? HHX !^««y N04X ^»V» 2asv *<$>*&* xww v^ifcrt: ♦run va^ VSA2 *<&tf* MHO ^>^V3 HMI^ ^<£AV >wcK75VAifi%^^ v^y^ yj>^y ♦Max [K>WVA2S»<?^ b?2K*(S f<?y^^ ni4H HXM52AV^»^ *S«aS> >^^^ M4ID -CT01> AVK? iP*%W ^i?^ ^iC^SC IDMI 15 Steven E. Cull Inane Study of . October 16, 1983- Th» 2\± two -colored 23t2i2 aiibufl "below represent the element b of the octahedral group 0, which la viewed as acting in the nano way on each of the 3 flubcubea of any given 2x£x2 cube. The arrangement of the 2lj. colored. cube a may be of some interest for Its combinatorial properties. "\ /'■ ■■■' A / ■ ■ \ / /..... V... ./. .-.. . .v.. / ■■■- . ■ L_\' ^ ._'.V_._ ■ v ■A." -..■ V 16 an fl. Culllrtarte Compound groups. Problem, November 6, l?83i Definition: Suppose a finite group G of order n cm b* represented as a group of permitatlons p., p 2 , • *■• Pn OQ m objects* where m< n. Suppose further that we can take th«se m objects to be distinct elements g,, Spi • -•* E^ *»f Q la such 4 *ray that the n products ■ • (p n ts 1 J)(p a (« 2 )J ... (Pnt^n are all distinct^ i.e. constitute all of G. If s'.icji a group G, confounded In this way from a of Its olements, exists, we call Q a compound group. Problem: Which (If any) finite groups are compound? 17 Steven S. Cull Inane Group compounda* Problem. November 10 i 1983- Definition: Let P be s group of n permutations on a finltw m- el anient get X., 1st G be a oaQtlpl i cativo group , end let f map I to O, Let r~{p i l^l^ol , let X = (k.: 1£ l±n\ . Derino ^sF^Obj ^(p x )= t ~jT^ r ( Pi ( x }). Tiio etructure [?, Bj f t tp } la a group compound. Problem; (1) Let Q be given. Fop which P and f in f[?) a coaet of some subgroup of G7 When is ^ a durjectlon? f2) Let P be glYen.« For which (J and f do Inverse images under <J> form a coast decomposition of FT When is ^ an. injection? 18 Steven, E. Cull Inane Table group a. . Froblam. NtfTercbor 27* 19fl3» Ha regard the (unborderod) tables of groups of order n ta men irraya over the symbol* 1,? n In *hioh tb* first row (read from left to right ) 1b the some no the first column (read froa top to bottom ). (The entry at top left represents the Identity but need not bo the ;:y:-t ol i.l Thus na regard eaeb of the arrays 1 £ 3 U 3 1 z «, if l tt 3 And 1H2 ad a table or the four-group. 3 1(12 2 £ 3 1 UM Ul 3 Such a table la determined (since It La a Latin square] by the entries lying below th* first row and to the right of the first coluim. Call this (n-l)x(D"l) portion of a group table an n-hox. (Note that the number B of n-b cocoa id In general greater than nllf, where N in tbe number of noniaorcirphlo group* of order n. For instance, forn-Uwehav* H^2, but B= Ci|I>(U> ratbar than (b,n(2).) For a given n M we may be able to sea something of how t.i.o Taricua order-n groups are interrelated by studying group actions on n-bojtes. Definition: Let 0(n} denote the direct product of (n-l)^ copies of Sq and regard tko components ot an el en ant e of 0(nj a a arranged In a □ (n-UxTn-l) array. Such .in element g acta COMponenUwlae la the obvious way on an n-boj to jleld art array that may or T»y not be an n-box. Suppose there exists aoiue- subgroup T of G(n) such that T la transitive on some sat B{T) of n-boKes that includes H n- boxes representing the N distinct (i.e., pairwise non Isomorphic) groups of order n« [We do not require that B(T) be closed under the action of T, nor even that each T- image of a member of B(t) be an n-box, ) We call such ft T a table group for a. Clearly for aaeb n there Is at least one table group T, namely G(n). That smaller T'a nay exist is shown by the following. Example: A table group for Jf is generated by the following elemsnta of ij(ij ), I (12) (IS) (13) (13) (13) (It) tite Ok) (12) (12) (13) (13) I (13) (14) tit) (Ik) 023 (12) 112) (13) (13) (13) lm W r Problem; What is the order of a sn.nl 1a at table group Tor n? la there some way to construct such group a that does not require knowledge of HI (The oasa for n a prima power seems of particular interest.) 41 Tha example is of course not a smallest table group for U, but is shown for its structural in to rest. 19 St von H- Cullinone - - Linear operat or-a In geonmtrle function apace* . Probloja. Jan. 5» 1?3U- Let X to a 2n-4ltaatialonil llrmar apace over ft f l&ld IT. Lafc ft map o tnke *»oh subepaofl 5 of X to a. function f £ : X ■ »■ K that la noiisaro on S and zero els* whore. [3ere fj 1j t *ort of char act oris tic function repreaenttng tha fl^ibspnoe 3, } Denote by P = P<£n, K, o) the linear Space o»er K spanned by the function f^. 1**9 call F a aBomatr^o funotlem apace. k Theorem; There la at least one P(2n, It, c} for whloh there exlata A iineax operator T, ncLing oil F, auth that T r taltsa the 1 -dlraan^ lon.il auhipacon of X (i.o., fun at ions fa representing such aubspacnaj to distinct r*l -dimensional gubspacoa of X, for l^r£3m-£. Proof; The matrix V at rdght r ?j :.■:- 1 ■:.■:• 3itJ SUoh alt Operator wbetl X la the linear 1;- apace ovor {jF{£), th» two-elftmeat Galois field. X: T: titiesta | l | l i l i l 4 e ft 6 1 li I It 1 i " C- c £ #41 i a d t I m n M I t 11 UHUDI I 4aM< 9D I I 0. j D 1 O 1 I i a J I Q I e o D I 4 9 I ]■ n 1 a ft i a « 1 a 1 I o O I e o l e> ■ ceo etc c id J etc v l 1 d ft 8 Q O DIB I I □ at oca 4 a hS tft ISO -•• a i Cog o L o « a i 4 a e e 4 a 4 g t a t a 6 C I D 1 o o ado boa e o 1 »o e i a I 4 DO 1 9 O SO) a I til 4 4 4 C | 4 04 4 4 4 o O I ft O-O od « a 40 4 D cool a a a a I ItC note o a o a 04SI 00 I o 1 a » l Olio a 44 e « o a a i o a 1114 ProMem: For what other apaeoa F{2n, K, c) does &uah a T oxlatT 20 Steven H, Cull inane Mamcmda and whirls . Expository note. Sept. 1*J, 198tj_. Modulo col or- interchange, and. rotations, thora are exactly 2. ways [aee fig. 2) to color the 6 faces of a cube ao that (a) each face la split diagonally into a black half and. a white half, and (d) there are exactly (j distinct iaafiea of the colored cute under tha group of £l|. rotational wyOTOfitriea of the cube. The rotational aynaastrioa of eacb anch. coloring form an ardor-S subgroup of leaving invariant an innoribed hexagon as In fig. 1. ThLn subgroup of oonaiats of the identity, potations of 120 and 2^0 degrees about a diagonal of the cube, and lQO-degraa rotations about each of 3 axoa Joining midpointa of opposite adgse of the cube. Big.. 1 Fig. 3 "Diamond" and "whirl™ cubos Identical copies of these cubes, variously oriented, can be assembled into larger cubical patterns with remarkable symmetry properties m, Pig- 3 A: Eight diamond cubes B: Sight whirl cubes Patterns A and B in fig. 3 yield a number of other symmetric pattern a when, their sub cubes are permuted (without rotation} aa follows. Let Sij_ act on the Ij. 1x1*2 "bricks" in each of the 3 partitions above; thB group A so generated can he shown to be triply transitive, of order l3l|ij-» and isoaiorphio to the affine group on the linear 3-apaoo over the two -element finite field* THEOHiS^; Patterns A and B each have 168 Images under A„ Each of theae images has some nontrivial symmetry (ordinary symmetry for A-imagea, ordinary or color- interchange symmetry Tor B-inages) under at leant one of a group of ^ rigid motions of the cube, © i<y8j + sac 21 atflven H. Cullinaiuo Affln© groups on flnaH binary spaces, EipoBitoiry note, Septi 2.$, 1?9L|., 1 Thaorom: Hi* afflne group ASL(3j2) AGL££,2) AGL(6,2) or order 322,560 319,979,530 1,290,157,^,^ is generated by a acting on partitions A A, B, G A, ^ 3 A, B, C, 2, 3 A, Bj Cj 1, 2-i 3> 22 Steven R. Culllnajne Visualizing fjL(2jp). Expositor? noto. Karch ?6, 19o?. "The typical Axiucple of a finite group la 0L(n,q}, tJiB general linear jp'oup of n a linens iond over fhft field, with 9. elenjertts, 11 -- J. L. Alpevifl. et> # a (ii) &) (r:j $} ill ^ : « arj es) (ft) {£) C*8 **— *■ 00 tit) ap (n) so j* — *■ '1 (S) UO (!?) W) { tf) (S) (f£) W (to) \i P ) J j W) (*) W 80 4 »- CSEJ Q S tii) s) (a) Ci) (?:) (K) & m \m 29 & (SI) IT!) 52 6? (ii) (9) )K '# (i?) tfl ^ K {it.} \Lh) m t5d &.) (;:) £2 Cli) (2) ea Tha 1^8- actions of GLi3,33 °rt a 3*3 eoordinate-nrray A ara 1 llufltrotcd abdv*, Tho matrices shown right -multiply the slomnnts of A, wh«™ (1[|1 flilMl-t , Actiona of GL{2,p) on a pjcp co ordinal ft -array have, the sama aorts of symnotri'Sa, *h©ra p 1 * <iny odd prima. 23 Stovso H, ffulllnane OUgjj) aatl&na Ott a cube. Expository not*. April 5, I9QS. Tbo 1$ diagram* halon illuatrat* aomo 97101110 tr la ■ of ail2 t }1 action* on tha 8 r,anirtro vectors of tha linear 2-spoca evti tha 3-clQ»ent del Tba vectors ana vlawed as laballng vertlsaa of a cube (plctuj-od aer» with & Blight dlutortion, to avoid ovarlappJLng llnea}. Tha 41&grui9 may have aoma hourlHtlo Tallin for th* Stud* of groups ganwatod by mljilag 0L[£,3) eotlona with those of other groups. 24 5toV*Q B. CUlli P£ no Qrcmp aetlopa on partitioned Problem ana query* April 5p 196?. Two wayo of partitioning & 72-aet: B Definition; Let a be the ^roup of degree 72 generated by mixing (1) actions of tho of fine group AOL (2, 3) on the est of nine 2x2x2 oubea In partition A a (2) like actions of AGL(2, 3} on each of the eight 3*3 sections in H, (3) actions or AGL(3,2J on the oet of eight 3*3 sections In B, and {!(,) lite actions or ACL (3, 2) on each of the nine 2x2x2 cubes in A. Problem: What Is the order of 0? Query! Clearly many similar problems could be posed* What results or methods are known? (Hota: nmnj equivalent coordinate ayatems for the arfUw actions above ara available via natural nappinga of the respective linear apneas onto 313 <?r 2x2*2 arrajs.) 25 Steven. E. Culllnana Generating tho acta a generator. Expository note. April 28., 1905- <3 1 % *+l *£ x^l **+* TtVx VI m *=? cioin) tn>n i saic fi', 14 BO I t *1 I I f 000 1*1* Olo I ' I I oo Hlo IQOl IQl 1 lltl LIU ;s m §3 A Sing Of 7-cyelQ The linear k.- space A Until" nsap S^ on. L 3-l on GP[B) L over GF[2) ( = £ copies of 3 L ) S^ and Sg a-otln£ on row 1 heloif yield the Miracle Octaa Generator [3] : D MHBB a SBH'S B HHSH a HI3DQ s 3BBB a BEEH B BBac £ D 1 ■ ■ ■ □■£■ I -a P^ Ha ■° II Anart from it a use in studying the 759 octada of a Steiner aystem S{5jG,2l|_) — and henco tho Mathieu group W^ — tba Curtis NOG nicely llluatrates a natural correspondence C [Conwoll [2]j P> 7 1 totwaoa (a) tie 35 partitions of an 8-aet 8 (bus!i aH G3TC8] above, or Convrell's 8 "heptads") Into two k.- set a, and (b) the 35 partitions oJT E Into. Toiir parallel ftfrltte planes. Two of tho H-pcrtitions have a common refinement Into 2-sete iff the same Is true of tho corresponding L-partitiona. (Cameron [l]j p. 6G) Note that is particularly natural in Ton 1, and "that partitions 2-5 in each row have fl itni i a-p structures. 1. Cameron, P.J., F&r&llaltgraa of Complete Deslanaj Camb. V. Pr. 197 k, 2. Comrell, G,M. , The 3- apace PG{3j£) and Its group, Ann. of forth. 11 (191Q) 60-75. 3- GurtlSj H,T., A nair tiomnlnatnrial approach to K^, Math. Prflc. (Jamb. Phil, 3oc. 79 [1976 J 25-l|5. 26 Steven H. Dull inane. 3ymmetry invurlanee tinder %?- Expository note* Aug. 22, 19&5. The qulntuply transitive HatMeu group M^g might be expected to thoroughly Bcramble any neat pattern it acta on* However, recent work by R. T» Ourtia and. J. H. Conway [ll has the following remarkable consequence. Theorem! The set of 7 infinite plane patterns be loir la Invariant (modulo rigid mot tone of the plaUB, and col or -inter change) under Gurtl a- Conway BL- actions on the Lpt3 motifs shown as quad rant a. Mote that each pattern has nontrivial aynmiBtry, modulo color- interchange. (The motifs are ? of the 132 hoxads in an S(5*6#12] ingeniously constructed in tlJ •) REFERENCE 1* Curt la j R. T.j The Steiner system 2(5*^*12), the Wathiou group H- 2 s^ul the "kitten, " Computational Group Theory, &d. iHchael D. Atkinson, Academic Preaa, 19&ht 353- 3^9 • 27 3. H. CUU-INAKE 11/17/85. Pinltfl groups of the fiaiaa order are aoajotimoa related by a nontrlvial identity* Example: 4 a a b a Q e a b a a. a c b b b c a c c b a B * 9 » b 6 a a a b c ■!1 Fl b C O b b c 6 (i c c a a b I'i'o hare, V if, x, j, -a -'.[fi, ft, bj el , {DJ X*(y+a) — I1-7J+ (**=}+ ij and hence The dual identity I' obtained by Into ^changing * and * in (I) also holds, Such a atructur* •*» two group a joined together by a nontri^ial Identity «-- might ba called a "bridge." Are there Infinitely Many eorts flf bridges? r ass grateful to S. Comer for tha following reformulation of this rather vague question. Definitions : Let B ={(<?,#,- ]i ((/,*) and (0,-) are groups} . Por ft subrariaty V« H let A denote the set of identities holding in [(V) f«P all (G,*\,*}eV* Similarly, define A 4 . Par any set of identities A in the language for S let V(A ) denote the variety of all members of B that satisfy A « Call a variety V reducible. if V=V{i H )fl Vl& m ). Problem : Are there infinitely many irreducible eubvarletiea of B7 28 S. B. Efullijiano Dyn&mlo and algebraic ooopetlbility of groups. Deo. 11, 19fi£. (A) Observation — Mo&lsonorphlc order- a group a, each transitively '.permuting, the asuae n points, isay generate & group smaller than A . Example — The four group ana fr^ acting on the parties a of a square j gone rate Dk. (B) Observation — ; Hon Isomorphic crder-n groups ara sometimes related by a nontrivial Identity. Exaraplo ** i c- a b o e a b e ft. a e c b b b c a & c c b a o ii o n b c e o a b e a a b g s b b t * ft c a a b wtth x*(y+*J= <x-y)4- (x»i)+ * V i,y,a cta r a,b,c} probleais: (A) For which tn,k} are there k nonlaomorphic order-n groups Gj Leach nlth the sans elements and the same Identity element ) and regular permutation representation!! i\ such that Kfat^K.^.r^^j)^!^! i (B) For which tn,lc) are there k nco is amorphic order-n Rreupe G# (each with the same elements and the same identity element) all Interrelated by a nontrlviel algebraic identity? {") For which {n,k} are there aolutlona to both {A) and (B)? 29 S. H, Cullinan& 0*On*trj of partitions II . Problems. Jaiiuary 11, 19flo, Pcflnlt^Qftg : liiTon 0* attR, and a finite (or countably infinite) aequance i, =(o-, a ? , »..) of poaitlTe real mtttiora such that Sa ; = t (op flueh that the partial 3UIsa Of K COUTSTge to a), call i o- partition ofa. Let L(0 b* tho following eurfaoe; LU ) - lUjjW^l** —* (**>*- TE.W*;Y* J. Thus L is i napping that lata Ua represent partitions by aurfoeea. (If the partial sums of t dlTarge but the o or ro spending eurf&oee converts, one might da fins L( E. } to be the Halt aurfaoa.) Theorem (Hieomaehua-Baohpt) i The 4nirfnco:i L((l, 2, .,., n) } all Interaeot at (1,2,3). Problems i 1. Do any other "natural" Tamil lea of part it lone yield Intersection theorems of a noutrlviel nature? 2. Kott do families of Infinite- aerie a partitions behave undei< J,? (Pop example, £ s = (1 t 2~ r ..., n ,-..}, Tor s>l.) 3. la the generalisation of L by taking (*,y,z)« C 1 l^.OnslMT difficult? 30 S. H. OullInenE Inacapea ill: PG(g,lj.) from F3(3,2). Expository note. Feb. l^ 1986. ■Thie note suggests a way to visualize the finite geaisetriaa recently described by A. fieutelspachwr in an excellent expository article Lll • Noto.-fei.oil -- hexadecimal characters for the 15 points Of FG(3,2)l 1 ~ 0OQ1 If = 0100 7 = 0111 A * 1010 8 sb 1101 2 = UG10 5 = 0101 8 = 1000 B a 1011 E = HID 3 = 0011 6 = 0110 9 - 1001 3 = 110O F ~ 1111 . Pacta about P5(3,2h the projective 3-spaee over GF(2) ; (A J Bsch of the 15 points the.j bs exproased a.a a sum of a unique pair of points from the set 3 - ■{l,2„3ti|-»6,C] . (B) Fifteen of tlw 35" lino a of FG(3,2) are distinguished by the fact that their pointc arise from partitions of s of the form 2 + 2+2- e.g., 3 =- il J 27Uf3,l l .1tfia,cj ylalcla the line i 3,^j7 1 = £1+2, 6+C, 3<-k} . (The remaining ?0 lines arise from petitions of S of the form. 3 + 3, "by summing pairs in the 3-aota.J to; Si* apreada, eaoh consisting of 5 mutually skew (i.e., disjoint) lines, asm be f craned from the 15 dlBtlnguiahed linos in (B). Zheae faote can be expressed graphically as follows. _l — 1 1 1 1 1 1 Lj — L 1 % 3 SI II" I z 1 IcD 18S Iqe jSQti 5PO = D E F w Points of PG-13J3J. (Note eymmetrlo- difference iwaa . J □Ens □BC3S iqe sac SftF 56S 4TF ft*A (B) 15 distinguished lines. {Celled an "Inscape" bo cause of part- whole relationship,) Ears D 1HT *H6 Mb H£ ED en annn The & aprestfa In (BJ . tlTote correspondence with S in {A) . } Beutelepacher describes a construction of KJ[2,lj.) with 21 points _ the 6 polnta of S and the 15 distinguished lines {BJ, am! 21 llnee — the 6 spreads (0) and the 15 point-psira (A), nSFmtSHCE 1. EeutalBpacher, A., 21 - 6-15: A connection between two distinguished geoaetrioa, Am. Math. Monthly 93 (Jan. '36) , 29-ia. 31 The relativity proDlem in finite fceamatry. Steb. SO, 19S6- This ia the relativity problem: to fix 00 jeetiijaly a class or equivalent coordlnatiatttions and to aa certain the group of t pans format iona S modi at log between thera. — II. Weyl, The ClaHsical Groups, Princeton Dniv* Pr*, 191^6, p. 16 In finite geometry "points" are often defined aa ordered n-tuplea of elements of a flnita {1.&,, Galois) field OF(q). What gaamotTla stmoturea ("frames of reference," in Weyl's tenpa) are ooordinatised by ouch n-tuploa? Weyl' a use of "objectively" seems to nioan that such structures should have oertaln objective — i«e,j purely geometric — properties invariant undai 1 each S, IMS note suggests suob a frame of reference far the affine Jj.-spa.oe QTor GFT2), and a ol&ea of 32£»560 equivalent eoordinatizatione of the frame. The frame : 4 )|-rl | array* The invariant atrnctu ra : Hie following set of 15 partitions of the frame into two S-seto, □nun ^3 Bd Sal U A representative coopdinatleB.fclon : 0000 oooi 0010 0011 0100 0101 0110 0111 zooo iaoi laio ion 1100 1101 1110 1111 TOie group : The group AGL(i]_,2) of 322, [JtO regular affine fcra-na fox-mat i one of the ordered J4- tuples over OP(£). 32 £, H. Cull Inane Gt-emp topologies. ProblopiB, March 31, 19o"o* If a group acta on a sot X f ther* is * natural olosure operation on subsets of X: define topological closure as cloFtura under G-actltsns, Then tha closed sot a (In both ana sea) ar* tha *!rpty sot, the G-orblfcs, and arbitrary unions of G-arMfcs, (A^ X is open iff A Is closed.) The result is a group topology T(O f X)» ttnTorfctinately, T{<J,X) la trivial if the group action, is tranaltive* But acta on th* power cot P(X) as well as on X, and *e nave X is nonasnpfcj' ** T(G,PU)> i" not trivial, and the --action la nontriviel e* T{G,*(X}) la. not discrete and not Tl {i.e., not all singletons ni* closed). (That a topology ia not ^ is unfortunate if the underlying net 1b infinite, but vary fortunate if tha underlying eat t« finite.} Let P =X, P n - FCF^tX)). and lot T a = T<G.r n U)) . Problem*: {1} Is thare a puraly set~ theoretic characterisation of the finite T n (i.e., among all oth*r topologies on P Q baHod on partitions that refine too cardinality partition)? {S} Consider tha topclogi** T Q Tor a faithful action P of G on X, (fi) la P alwaya determined by T , T^., ... r T n for a °me n= n{P)T (b) If H<G, how are thii T. for H related to tha T ± foT Gf it) ir x is ccuntabXy infinite, cun m regard tha minimal eloaed sots of f- as "natural" G-orbita on *qb» oontinuumt 33 S> Ht Culllwmo Picturing the arefllloat projective 3-flpa.c*. April 26, 198 6 > □mno DBoon BHfflB HSffflH BfflliB Tha IS points ]HBH Lsfl L_J kjJ P_5 HGHH QDDC The & spreads in A QCHH SBBB raaBQ The 25 hyperplanes nnnn DHHH The £ figures at left ahow a GjTHplactle polarity a; *ach point liefl in its oorro spend- ing hyperplane. The 15 linos fixed undar * are shown in fig. A below. CDDD DBGH HBO □□as nnEH B The 35 lines moo n»nra HDD Sums of the i'-subsctu of A pictured in A anon DHBn DBBH □ BBS Sums qT the I4.- sub sot 3 of A pictured In B or C CDDD DBBS Df?aa DDBB The K- T. Curtis oorre spends nee bo t ween the 35 lines and the 2- Subsets and 3-&ybsota of a &-aat. This underlies IToj. . 34 A lLti«di* aasepi.** rslatad tn W-,| . Ma 7 ft, l'>"t, Wgur*j A.B^C BhBW tb* 3? linen or FQ(J,£); tig. A la ■ llnau* ocnple*. DSSH HEupm cnbn □HOJH nasa naaE nnna DHEH □□GB DSHD HHGH □HUB saee DDDD DEHQ DQEH DEQB EHBH HELBa HEIDI] B + C Ft put* a JL', R p ,-3' show and tbB 3 ^ partitions una eyas esse QISl ODDD □nnn onnn nnnn a. + a til* B.Tt. Curtis oorrenp of mi 3-4* t lfito t«s li A 4 C ondono* between th* 35 UnM *#ts a Thin undorlioa ^2h' B'+C EQDD QIBQ DB0B DIBH IBIB BBBB BBBB A' + B 1 CDDD DBHQ DHSE DBBS HEED BEEE Llll A' + C 35 S. H. Cullinane The 2-cubseta of a 6-aat are tlie polnta of a FG(3,2). Mstj 2e, 1SB6, This nota Tms suggested by (l) A. Beutelapaoiuep'e model [13 of the 15 point a of FG(3»2) as tins 15 partitions of a 6- set into throe Z-natu, and by (£) H, T» Curtis 1 a model E3J of the Conwell aorreapcKiaBnco C21 Tjatwaen tha 3$ linso of P5(3»a) and the 35 partitions of &n 6-aet into two Ti-aBta. If X ia a finite aet, we may regard the powor set FtX) as en elementary abelian 2-groTip in which addition is the sofc-thBoretic Bymmetrlo-dlffepwiQi* operation. Let K(X) be the subgroup of P(X) oonaistizi^ of and X, and let QtX3 = P(X}/k{X1. ffhen X ia a 6-set , the Z-aubsetfl form a subgroup A. of <i(X) T.'hosa noniaro dements we may take aa the pointa of a ^3(3,2), with 00 1 linearity defined in the obTioua way. □SHE S§i§ SEEK A 1 A subgroup of Q{X] Illustrating Subsets of a illustrating {l) the 15 2-subsot3 of a 6-set £1) the Ourtis correspondence between 12J the 15 points of PG(3,£> A-to) and the 15 partitions of a 6- set into three 2— Bete [2) a linear complex Ln FG-OpS) HEFEEtESCES 1. Bflutelspaoher, A., £1 - o = l5': A connection between two distinguished goonetpisa, A™er. ifcth. Monthly 93 (1966) 29-l|l . 2. Conwell, (I, M., The 3-apaco PG-U.2) and Its croup, Ann. of Math. 11 (1910) 60 -76 (eap. p. 72). 3. Curtis, R. T._- A new combinatorial approach to IjUt, Kith. Proc. COM), Phil, Sac. 79 (1976) 25-h2. 36 s, r. cigiTmne 21 projective partitions. Research nato* June 6 4 1986. Shown below are the Zl -point projective plane PG{£»fy) and Its dual. The point | (or lines) ere the 21 partitions of a 6-3et into disjoint seta A, B T w3iere |AJ - S or 1* Lines (or points) t QQBaHQ HHQBEQ HHESEH 5B§ Paints (or lines): caaQEis seeesq ssbhss BgB Foists on the ahwe Unee (or lines on the above points) : 0300012 BHQQQQ BSQDnQ ESQ EBEHSia QQBBE3Q QEESQQ QBQ Bssasa aasses assess eas HSBBas gSGi@g SHSHBffl EBB SB@§BB BOSHES SHSBQH 00H The 6-aet pemmtatlon interchanging points and line a is ftom the lOacle Getad Generator of B. T. Curtis [1, p, 23] * REFERENCE 1. Curtlfl, R. T-, A now o-ornhinstorial approach to M^, Hath. Frqc. Camh. Phil. 3oe. 79 (1976), 2S-UZ, 37 S, II. CUHinattB An outer automorphism of S4 related to Vgj,, June 11, 1986. Figure A below shoirs the 3 - subs* t 3 of a S-S&t S; figure B a hows tha locutions in A of tho triples of £-aubsata that partition £. nnmm DSSS ^ e anen ^™ ! QHBH eJCJEJU ddbo ssnn A B ■ Together j A and B specify a a or re spontanea C between tha 15 subset a and the 15 part It ion a* This correspondence 1 a : 1 m l* In a natural way to (l) n model of the proJacliTe piano FOC^A) In which. Che 21 points {and slao the 21 lines) ire the 21 partitions of 5 Into Hubseta a. a. -i uere 1X1 = £ or 1; (2} the Gonwell mapping of the 35 Uj. V W -partitions nf an d-aat onto the 35 lines of FO(3j3), which preserves certain Intersection properties; (3} the Rh T. Curtis "MOO" Bedel of tha Stelner ajatam fll5,{},2li.J afld of StaN as the model ■ s automorphism group. 1st fia^a' exchange rows 2 and 3 In sack 3*2 picture a In A, una i«t c r nap a 2- subset s to C(s<)< If w& regard tha 2-a*ta and partition* a a transpoaitlona and products of transpositions, c 1 JUndUcea aa outer automorphism p of S;. {In the above PG^i).), 5g and p(s^) act in concern aa a jstroup of eolllneatlons. ) 38 &„ H, Gull inane Picturing outer sut ORorphi ana of S&. Expository note. Jul? 3t 1986. DBCB anats HQBU DDEQ □ □00 D1QBQ □sqq □ HOD Iwaj l — i i/i 541 * H N 1 11 * Ills III1 ilii ill g M|]|n 9 ^ Shown above are two ways to picture sow outer automDrphlHms of Sfc that Irnvo b^an discussed Ln the literature (6 in tl), ^ In £2)). In the top row, figure X shows the 1$ 2-aete in a 6-flet S, and 0,£ show the locations In X of triples of 2- acts that partition S. The second row shown the c-QTreap ending permutations. Each row's & ,$ contain & special 5~ m bsets : In the top i*ow theae f-subaeta are apreada of Unas Ln ft FG(3»«J); in the second row they are p&ralloliaiaa of S. Suoh S-mibaata teach of which can be selected in 6 waja, then nrranged in $1 ttnyo) determine the »J outer automorphlstaa of sj. [1J Conway, J* 8., Three lecture on e*captlonol groups (section 2.3K in Finite Simple (Jroupa, ad, Fowell and Higtcan, Academic Pr., 1971 1 (2) J&nuaE, a., and Restrain, J., Outer automorphisMS of S&, toer. Math. Hcnthlj 39 (June- July 1982) , h.Q7-iao. 39 3. E* Culllnane in a cap e a IV: Inner and outer group actions. July 11, 1S86. Thla nota TPB.B suggeated by J. E. Conway's const ruction (1) of an onder*2 outer automorphism of a^. 11 1 1 llli x 1 1 1 1 *X mm dhqci ii HHEE Figures A and B above each show 16 pemnutationB of a 16-aet that generate groups G{A) and G(B), r a ape. c timely. Figure X shows 16 subsets of a 3.6-set, The group a G-(a) and Gr(B) can act on figure X In two ways: by an Inner action on &ach of the 16 Ipta part a Indlrldttftlly, or by an outer action permuting the 16 parts. Theorem: Let a denote any permutation In A, and, let t denote the permutation in the corps spending location In B. Then the inner ( outer) action of a on X induce b (is induced by) the outer tinner) action of b on X. The group G[A), antJ henoe 3(B), is isomorphic to Sy, and tbe map taking each a to its corresponding b extends to an involutiva outer automorphi sm of St. REFERENCE (1) Conway, J- II., Three lectures on exceptional groups (eection 2,3), in Finite Simple Group a, ed» M. B. Powell and G. Hlgja&n, Academic Preas, 1971, 40